Theorem eqelssd 3156

Description: Equality deduction from subclass relationship and membership. (Contributed by AV, 21-Aug-2022.)

Hypotheses

Ref	Expression
eqelssd.1	|- ( ph -> A C_ B )
eqelssd.2	|- ( ( ph /\ x e. B ) -> x e. A )

Assertion

Ref	Expression
eqelssd	|- ( ph -> A = B )

Distinct variable groups:   x, A   x, B   ph, x

Proof of Theorem eqelssd

Step	Hyp	Ref	Expression
1		eqelssd.1	. 2 |- ( ph -> A C_ B )
2		eqelssd.2	. . . 4 |- ( ( ph /\ x e. B ) -> x e. A )
3	2	ex 114	. . 3 |- ( ph -> ( x e. B -> x e. A ) )
4	3	ssrdv 3143	. 2 |- ( ph -> B C_ A )
5	1, 4	eqssd 3154	1 |- ( ph -> A = B )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1342   e. wcel 2135   C_ wss 3111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-11 1493  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-in 3117  df-ss 3124

This theorem is referenced by:  fiuni  6934  ennnfonelemrn  12289  ennnfonelemdm  12290  unirnblps  12963  unirnbl  12964  dvidlemap  13201  dviaddf  13210  dvimulf  13211  dvcj  13214  dvrecap  13218